Gerbes have now found their way into the stacks project. I am still working on this, but some basic material is now there.

It was a little bit more complicated than the discussion here. The reason is that you want to say what it means for one algebraic stack to be a gerbe over another. I decided not to give a geometric characterization as the definition, but rather to define what it means for one stack in groupoids over a site to be a gerbe over another. This is done in Stacks, Section Tag 06NY. We then discuss what this really means for a morphism of algebraic stacks in Morphisms of Stacks, Section Tag 06QB.

It turns out that the “topological/stack-theoretic” definition does entail that if an algebraic stack X is a gerbe over the algebraic stack Y then X —> Y is flat and locally of finite presentation and is fppf locally over Y of the form [Y/G] for some flat and locally finitely presented group algebraic space G. In fact this characterizes gerbes — i.e., we could have defined them this way. See Lemma Tag 06QH and Lemma Tag 06QI.

We say that an algebraic stack X is a gerbe if it is a gerbe over some algebraic space. Similarly to the above it turns out that this happens if and only if the inertia of X is flat and locally of finite presentation over X, see Proposition Tag 06QJ.

Let f : X —> Y be a monomorphism of algebraic spaces. Is f representable (by schemes)? After hitting this with a bunch of standard arguments I was led to the following commutative algebra question:

Question: Let A —> B be a local homomorphism of local rings such that the two maps B —> B ⊗_A B are essentially etale and such that A is their equalizer. Then is the map A —> B essentially etale?

This is a first approximation; I have been unable to find an exact translation of the problem on monomorphisms into algebra. The answer to the question is (I think) yes if A is a local ring of dimension 0, or if A —> B is flat (descent of \’etale ring maps).

By the way, I should mention that the statement on monomorphisms of algebraic spaces is true when the morphism is locally of finite type. Namely, any separated, locally quasi-finite morphism of algebraic spaces is representable (by schemes), see Lemma Tag 0418.

In the stacks project a Deligne-Mumford stack is an algebraic stack X such that there exists a scheme U and a surjective etale morphism U —> X. An algebraic stack X is said to be DM if the diagonal Δ : X —> X x X is unramified. In fact Theorem Tag 06N3 says:

X is DM if and only if X is Deligne-Mumford.

An algebraic stack X is said to be quasi-DM if the diagonal Δ : X —> X x X is locally quasi-finite. The analogue of the theorem above is Theorem Tag 06MF which says:

X is quasi-DM if and only if there exists a scheme U and a surjective, flat, locally finitely presented, and locally quasi-finite morphism U —> X.

The proofs of these theorems are completely parallel. Assume X is DM (resp. quasi-DM). We try to construct etale (resp. loc fp + flat + loc quasi-finite) maps from schemes toward X. In both cases the strategy is the following:

1. Pick a smooth morphism U —> X,
2. choose a suitable point x of X,
3. let F be the fibre of U over x, and
4. “slice” U, i.e., find a complete intersection V(f_1, …, f_d) ⊂ U such that f_1, …, f_d form a regular system of parameters (resp. regular sequence) at some point of F

In both cases the proof shows that, after possibly shrinking U, the morphism V(f_1, …, f_d) —> X is flat, locally finitely presented, and unramified (resp. locally quasi-finite). A bit of care is needed in choosing the point x on X. I decided to use “finite type points”; in both cases one then has to do a bit of work to show that the “fibre F” has desirable properties: in the DM case one need to produce x such that F —> U is unramified and in the quasi-DM case such that F —> U is locally quasi-finite.

The reasoning above is completely standard. However, there is a way to deduce the first theorem from the second. I decided against arguing like this in the stacks project as it is perhaps a little nonstandard. Here is the argument. Let X be DM. By the second theorem we can find U —> X which is surjective, flat, locally of finite presentation, and locally quasi-finite. Let H_{d, lci}(U/X) be the LCI locus in the relative degree d Hilbert stack of U over X (see Section Tag 06CJ). Then H_{d, lci}(U/X) —> X is smooth (this is explained in the proof of Theorem Tag 06DC). But of course it is clear that H_{d, lci}(U/X) —> X has relative dimension 0, hence it is etale. This doesn’t quite finish the proof because H_{d, lci}(U/X) is (as defined in the stacks project) an algebraic stack and not an algebraic space; but a straightforward argument shows (because X is DM) that the disjoint union for varying d of the open substacks of H_{d, lci}(U/X) having trivial inertia surjects onto X.

This morning I introduced a notion of residual gerbe for a point x on an algebraic stack X. See the (currently) last section in the chapter Properties of Algebraic Stacks. I decided that the residual gerbe of X at x should be a reduced, locally Noetherian algebraic stack Z whose underlying space |Z| is a singleton which comes with a monomorphism Z —> X such that the unique point of Z maps to x.

In the generality of the stacks project I cannot show that residual gerbes always exist. If a residual gerbe Z does exist, then it is unique. In fact, it turns out that there exists a field and a surjective, flat, locally finitely presented morphism z : Spec(k) —> Z (which is a very convenient property to have because we work in the fppf topology). For any algebraic stacks there are alway points where the residual gerbe does exist, namely the points of finite type.

In Appendix B of the preprint “Etale devissage, descent and push-outs of algebraic stacks” David Rydh has shown (I think) that residual gerbes (as defined above) exist for any point of a quasi-separated algebraic stack (his results are actually stronger). This implies that the definition above does not conflict with the definition in the book by Laumon and Moret-Bailley.

One curiosity is that we haven’t yet defined gerbes in the stacks project. The fact that a residual gerbe “is” a gerbe (over something) isn’t yet documented in the stacks project.

Let me know if you have any comments or suggestions.

Alex Perry wrote a chapter on formal deformation theory for the stacks project following Schlessinger and Rim. Please read the introduction of that chapter for more information.

I intend to work on this chapter a little bit more in the near future in order to allow for finite residue field extensions (i.e., work with Λ —> k of finite type). The way the chapter is written however, I believe only minor changes will have to be made.

Once this is done we intend to use this material to study the formal local structure of algebraic stacks and to explain Artin’s criteria for Algebraic Stacks. One big obstruction looming in the future is the general Neron desingularization (Popescu). I’m not yet sure how to deal with this.

More immediately what we really need now is a couple of examples where the theory applies directly as written up. Alex and I listed a few obvious examples at the end of the chapter. If you feel like writing one of these up (should not be more than a few pages) using the framework we have in place please email me (so we don’t do double work).

In this post I claimed that a formally smooth ring map has a cotangent complex which is quasi-isomorphic to a projective module sitting in degree 0. I thought this was in Illusie’s thesis. But when Wansu Kim asked me for a reference, and when I tried to find it today, I couldn’t find it.

Now I think it is simply wrong! I constructed what I think is a counter example and put it in the chapter on examples (search for cotangent complex). Let me know if I made a mistake… again.

The stacks project now contains Artin’s Theorem Tag 06DC:

Let f : X —> Y be a 1-morphism of stacks in groupoids on (Sch/S)_{fppf}. Assume that X is representable by an algebraic space, f is representable by algebraic spaces, surjective, locally of finite presentation, and flat. Then Y is an algebraic stack.

Extremely loosely speaking this means that to verify that a stack in groupoids Y is an algebraic stack, it suffices to find a flat cover by scheme, smoothness not required!

This has some pleasing consequences which we have not yet spelled out in the stacks project (most of these are similar to the consequences to the corresponding result for algebraic spaces, see Theorem Tag 04S6 ff). For example

1. Given a stack in groupoids X over a scheme S and an fppf covering {S_i —> S} such that X|_{S_i} is an algebraic stack for each i, then X is an algebraic stack.
2. Given a flat, finitely presented group scheme G over S acting on a scheme X over S, then the quotient stack [X/G] is algebraic.
3. Given a groupoid scheme (U, R, s, t, c) with s, t flat and locally of finite presentation, then the quotient stack [U/R] is an algebraic stack.

And so on and so forth. Moreover, in the process of proving the theorem stated above we proved some results on algebraicity of spaces of sections, relative morphisms, restriction of scalars, and finite Hilbert stacks, most of which can now be considerably improved.

Our next goal in the stacks project is to add more basic theory on algebraic stacks, add some material on deformation theory, and (perhaps) something on approximation theory.

A fun fact is that the graph of logical dependencies for Theorem Tag 06DC has depth 68 and has 4006 edges!

This is my second report on the second semester of a yearlong algebraic geometry course for first year graduate students here at Columbia University, based on Ravi Vakil’s lecture notes. The first report is here. Please visit Ravi’s Math 216 blog and find the complete set of lecture notes here.

Besides some minor and unimportant annoyances with the text, I found teaching out of these lecture notes very pleasant. What turned out to be, for me, a key feature of his notes is Ravi’s intent to do things in the correct generality:

We will work with as much generality as we need for most readers, and no more. In particular, we try to have hypotheses that are as general as possible without making proofs harder. The right hypotheses can make a proof easier, not harder, because one can remember how they get used.

As I worked my way through the material I felt that Ravi mostly succeeded in this and it gave me the confidence to be less general! For example, I proved a bunch of results on ample and very ample invertible sheaves in the course working only with morphisms between schemes of finite type over a field. I stuck with Noetherian integral schemes whilst defining the Weil divisor class group. I talked about effective Cartier divisors, but avoided talking about Cartier divisors (a horrrible invention IMHO). Etc, etc.

Doing this allowed me to cover more ground than I usually do in an algebraic geometry course. I was able to do pushforward and pullback of divisors for finite morphisms of regular curves and prove the “n = ∑ e_i f_i” formula if you know what I mean. I was able to introduce cohomology for quasi-coherent sheaves on quasi-compact and separated schemes and actually prove some interesting theorems about it, by only doing Cech cohomology (this is probably the best time saving feature of the notes — it is one of those “why didn’t I think of that” things). Using this I was able to prove q-gr(A) = Coh(X) when X = Proj(A). A trivial consequence of these basic theorems is then the Riemann-Roch theorem in the form χ(X, L) = deg(L) + χ(X, O_X) on a projective regular curve X.

Finally, at the end I diverged from Ravi’s notes. I introduced dualizing sheaves for projective regular curves X by requiring Serre duality to hold for locally free sheaves (since there is only H^0 and H^1 you don’t need the cup product here nor Ext groups). I proved the existence of an invertible dualizing sheaf ω_X on X by first proving it for P^1 and then (using duality for a finite flat morphism) for any X by choosing a nonconstant rational functor. I defined the genus of a projective regular curve X over a field k, assuming H^0(X, O_X) = k, as g(X) = dim H^1(X, O_X). Then Serre duality gives deg(ω_X) = 2g(X) – 2. I proved that every genus 0 regular projective curve X is a conic. I proved that ω_X = Ω_X^1 if X is smooth over k (although here I had to assume something about a trace map on differentials). Finally, I explained how this leads to Riemann-Hurewitz using functoriality of differentials.

It may seem depressing to not be able to get much beyond RR and RH in a yearlong algebraic geometry course. But what really counts is for students to learn a whole language. Working through Ravi’s notes is a great way for students to do this. Thanks Ravi!

Let R be a local ring. Let J ⊂ R be an ideal generated by a Koszul-regular sequence. Let I ⊂ J be an ideal such that R/I is a perfect object of D(R) and such that R/J is a perfect object of D(R/I). Then, is it true that I and J/I are generated by Koszul-regular sequences in R and R/I?

In the Noetherian case you can just say “regular sequence” and the conditions just mean that I has finite projective dimension over R and R/J has finite projective dimension over R/I. But the way the question is formulated makes it believe-able that if the question has answer “yes” in the Noetherian case then the answer is yes in the general case. I have tried to prove this and I have tried to find counter examples, but I failed on both counts. I would appreciate any comments or suggestions.

Just an update on what’s been going on since the last update. The following list is roughly in chronological order.

1. Jonathan Wang send us a bunch of lemmas which help determine whether a given stack in groupoids is an algebraic stack.
2. We added enough material on finite Hilbert stacks so we can use them. These results are mainly contained in the chapter entitled “Criteria for representability”. It came as a big relief to me that these results are painless to prove given the results on algebraic spaces at our disposal.
3. Removed the ridiculous term “distilled” and replaced it by “quasi-DM” as suggested by Brian Conrad.
4. Started a chapter entitled “Quot and Hilbert Spaces” where we will eventually put results on existence (as algebraic spaces) of Quot spaces and Hilbert spaces. So far it only contains a discussion of “the locus where a morphism has property P”.
5. Added an example of a module which is a direct sum of countably many locally free modules of rank 1 but is not itself locally free.
6. Added a bunch more basic results on modules on algebraic spaces, and on morphisms of algebraic spaces.
7. The pullback of a flat module along a morphism of ringed topoi is flat. We only proved this in case the topoi have enough points. The general case (due to Deligne) is a bit harder to prove, and we’ll likely never use it.
8. The fppf topology is the topology generated by open coverings and finite locally free morphisms. Discussed previously on the blog.
9. Basics of flatness and morphisms of algebraic spaces (openness, criterion par fibre, etc).
10. Added an example of a formally etale nonflat ring map due to Brian Conrad.
11. Infinitesimal thickenings of algebraic spaces. We study these using the earlier results on algebraic spaces as locally ringed topoi discussed earlier on this blog. A key technical ingredient is that a first order thickening of an affine scheme in the category of algebraic spaces is an affine scheme. This can be tremendously generalized (see work by David Rydh), but that would require a _lot_ more work.
12. Universal first order thickenings for formally unramified morphisms of algebraic spaces.
13. Fixed section on formally etale morphisms of algebraic spaces.
14. Section on infinitesimal deformations of maps of algebraic spaces. This is now very slick, due to the work on thickenings above.
15. Fixed proof of relationship formally smooth morphisms of algebraic spaces and smooth morphisms of algebraic space.
16. Formal smoothness for algebraic spaces is etale local on the source.
17. Relative effective Cartier divisors.
18. Lots of material on regular sequences, regular immersions, relative regular immersions, all intended to be used eventually to define local complete intersection morphisms.
19. Introduced the following algebra notions:
    1. Pseudo-coherent complexes
    2. Tor amplitude and complexes of finite tor dimension
    3. Perfect complexes
    4. Relatively pseudo-coherent complexes
    5. Pseudo-coherent ring maps
    6. Perfect ring maps
20. Introduced the following types of morphisms of schemes:
    1. Pseudo-coherent morphisms of schemes
    2. Perfect morphisms of schemes
    3. Local complete intersection morphisms

Among some of the properties of these we proved that local complete intersection morphisms are fppf local on the target and syntomic local on the source. Hence it makes sense to say that a morphism of algebraic spaces is a local complete intersection morphism. We should now be in a good position to define the “lci-locus” in the Hilbert stack, which is our next goal.